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Home , Alessandro Strumia, Vacuum energy

PHYSICAL REVIEW D 101, 115002 (2020)

Relaxing the Higgs and its by living at the top of the potential

† Alessandro Strumia1,* and Daniele Teresi 1,2, 1Dipartimento di Fisica “E. Fermi”, Universit`a di Pisa, Largo Bruno Pontecorvo 3, I-56127 Pisa, Italy 2INFN, Sezione di Pisa, Largo Bruno Pontecorvo 3, I-56127 Pisa, Italy

(Received 17 February 2020; accepted 19 May 2020; published 1 June 2020)

We consider an ultralight scalar coupled to the Higgs in the presence of heavier new . In the electroweak broken phase the Higgs gives a tree-level contribution to the -scalar potential, while new physics contributes at loop level. Thereby, the theory has a cosmologically metastable phase where the light scalar is around the top of its potential, and the Higgs is a loop factor lighter than new physics. Such regions with precarious naturalness are anthropically and environmentally selected, as regions with heavier Higgs crunch quickly. We expect effects of rolling in the dark-energy . Furthermore, vacuum up to the weak scale can be canceled down to anthropically small values.

DOI: 10.1103/PhysRevD.101.115002

I. INTRODUCTION values of ϕ such that the Higgs mass is a loop factor lighter than new physics. Scalars with mass m below the Hubble scale H0 This observation is relevant for the Higgs-mass hierarchy can still lie away from the minimum of their potential and problem if, for some reason, ϕ lies close to the maximum of thereby undergo significant cosmological evolution at its potential. The authors of [7,8] explored possible reasons: present times. This can be of possible relevance for barring the possibility that ϕ is a ghost, they added extra understanding the apparently unnatural hierarchy of scales wiggles to Vϕ in order to generate local minima around the observed in Nature: much below the maximum, arguing that they can be favored cosmologically weak scale much below the Planck scale. See [1–12] for because they inflate more. some recent attempts of understanding hierarchies via We consider a simpler way of living at the top. Since the cosmological evolution. potential Vϕ is flat around its maximum, regions suffi- We here consider a scalar ϕ with the shift symmetry that ciently close to it can be cosmologically metastable, as long keeps its potential flat broken by small interactions to other as ϕ is so weakly coupled that its mass is small, m ≲ H0. fields, in particular to the Higgs doublet H and to other Instead, regions away from the maximum quickly roll heavier new physics that generates a Higgs mass hierarchy down and catastrophically collapse into a big crunch, as problem. The interaction with the Higgs can be para- illustrated in Fig. 1. In this way, only regions where the ϕ M2ðϕÞ metrized by a -dependent Higgs mass, h . The model Higgs is light are long-lived: a small Higgs mass is selected. will cosmologically generate a little hierarchy, maximal in ϕ The volume at late times gets dominated by such the special case where the couplings to new physics are near-critical regions at the border between the broken and mediated by the Higgs. unbroken Higgs phase. Indeed, integrating out the Higgs generates a tree-level We also find that this setting allows for a range of values ∼− 4ðϕÞ negative contribution Vϕ Mh to the effective poten- of the vacuum energy, up to the weak scale in the most ϕ 2ðϕÞ 0 tial for if Mh > , i.e., when electroweak symmetry optimistic case, thereby providing a mechanism (alternative breaking takes place. Heavier new physics contributes at to a landscape of vacua) for the anthropic selection of loop level. The full potential Vϕ can have a maximum for the cosmological constant down to the small observed value. *[email protected] This scenario generates a little hierarchy, making the † [email protected] weak scale one or two loop factors below the scale of new physics. The needed new physics is unspecified and might Published by the American Physical Society under the terms of be a full solution to hierarchy problems, such as super- the Creative Commons Attribution 4.0 International license. Further distribution of this must maintain attribution to symmetry (see, e.g., [8]), or possibly a landscape with the author(s) and the published article’s title, journal citation, electroweak, rather than meV, spacing (e.g., from string and DOI. Funded by SCOAP3. [13–15] or [16,17] theory).

2470-0010=2020=101(11)=115002(8) 115002-1 Published by the American Physical Society ALESSANDRO STRUMIA and DANIELE TERESI PHYS. REV. D 101, 115002 (2020)

couplings g.2 For example, the heavy new physics could be supersymmetric and ϕ could be a quasimodulus. We want to compute the effective potential for ϕ. First, we integrate out the heavy new physics obtaining the effective field theory around the weak scale, with effective potential

2ðϕÞ ð ϕÞ¼ − Mh j j2 þ λj j4 þ heavyðϕÞ ð Þ Veff H; V0 2 H H Vϕ ; 1

where λ ≈ 0.126 is the Higgs quartic. The potential contains the tree-level coupling of ϕ to the Higgs, parametrized by 2ðϕÞ FIG. 1. Effective potential for ϕ around its maximum. After Mh . The squared Higgs mass might receive unnaturally different patches of the have different values of large corrections from heavy new physics, but what matters ϕ. In most patches ϕ is away from its maximum and slides down here is its value renormalized around the weak scale. In leading to a crunch. Only patches where ϕ is near the top are agreement with the decoupling theorem, heavy new physics metastable on cosmological timescales. These correspond to a heavy can instead give the Vϕ ðϕÞ term in the effective potential Higgs mass one or two loop factors lighter than new physics. (to be computed at one- or two-loop level in Sec. II A). 2ðϕÞ 0 Finally, we integrate out the . If Mh > An observable consequence of precarious naturalness is (such that, in our notation, the electroweak symmetry is that in an Oð1Þ fraction of the long-lived regions of the broken), the Higgs gives a tree-level contribution to ðϕÞ Universe, the scalar field ϕ is rolling down the potential Veff . The Higgs also gives a loop-level contribution, now, with detectably large effects on the equation of state of the usual running of the cosmological constant, that can be . The other robust consequence is the presence neglected with respect to loop effects of heavier fields. We of new physics coupled to the Higgs around or below the thereby obtain the effective potential for the light field ϕ ≈20 TeV scale, in order to generate the maximum at the observed value. 4ðϕÞ Mh 2 heavy V ðϕÞ¼ 0 − θð ðϕÞÞ þ ðϕÞ ð Þ The paper is structured as follows. In Sec. II we present eff V 16λ Mh Vϕ ; 2 the model and study its cosmological dynamics in Sec. III, where we illustrate the mechanism leading to precarious θ ¼ 1 M2ðϕÞ > 0 naturalness. We then discuss bounds and signals in Sec. IV where for h and zero otherwise, so that the ðϕÞ ϕ 0 tree-level part of Veff is flat for < . and finally conclude in Sec. V. heavy We assume that the sign of Vϕ is such that, together ðϕÞ with the negative tree-level Higgs term, Veff has a local II. PRECARIOUS NATURALNESS: THE SETUP maximum. Without loss of generality, we can shift ϕ so that ϕ ¼ 0 We consider the (SM) with mass the maximum lies at . We next assume, for simplicity, scale M ≈ 125 GeV plus heavier new physics at the mass that the two terms can be approximated by first-order h Taylor expansions scale M ≫ Mh and an ultralight scalar ϕ with mass-scale m ≪ Mh. A Higgs-mass hierarchy problem arises if the Higgs 2ðϕÞ ≃ 2 þ ϕ heavy ≃ ϕ ð Þ Mh Mh0 g ;Vϕ gA : 3 interacts significantly with heavier new particles. We leave the heavy new physics unspecified. This includes the possibility of new physics, such as , that The parameter Mh0 must be negligibly different from the ϕ makes its scale M naturally smaller than the ultimate physical Higgs mass, 125 GeV, given that will lie very Λ close to the maximum. The maximum condition then cutoff of the theory at some scale , possibly around 2 1 ¼ 8λ the Planck mass. implies Mh0 A. As we now show, this is loop sup- On the other hand, the lightness of ϕ is naturally pressed with respect to M. understood if the ϕ potential is protected by a shift symmetry, broken at some scale ≲Λ by very small 2Such tiny couplings mediate a force much weaker than for the Higgs. However, this is not in contradiction with the weak gravity conjecture [18], since the force is attractive, so that its effect adds up to gravity [19]. Sometimes it is argued that 1The possibility of new physics at the weak scale that makes also scalar forces must be stronger than gravity [19], but the the Higgs mass natural would have been simpler, but it has been arguments for this are weaker [20] and disappear for fields with disfavored by collider searches. sizable gauge interactions, like the Higgs boson in our case.

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FIG. 2. Combined selection of a small cosmological constant, in addition to a small Higgs mass, for m ≪ H0. See the text for details.

V 2 A. The loop contribution to ϕ from heavy new physics M ∼ ð4πÞ Mh ∼ 20 TeV; ð6Þ If the full theory is valid up to some scale Λ ≫ M,a heavy having assumed ln Λ=M ∼ 1 and λ ∼ 1. Order one dominant contribution to V is computed by solving the HS ϕ couplings are unavoidably needed in models where new one-loop -group equations (RGE) in the physics at mass M provides a natural solution to the big heavy full theory. Otherwise Vϕ arises as threshold corrections hierarchy problem (for example in supersymmetry the at the scale M ≈ Λ. For the sake of illustration, we compute Higgs has top Yukawa couplings to sparticles). heavy the minimal contribution to Vϕ in a simple toy model: The less favorable possibility is that both H and new ϕ we assume that the new physics at scale M is just a singlet physics S directly couple to . We can parametrize the new 2 2 physics direct coupling as M ðϕÞ2 ≃ M2 − g ϕ,withthe scalar S with quartic coupling λHSS jHj to the Higgs. S S heavy ∼ To start, in order to compute the minimal V at two shift- coupling gS g. RGE running of ϕ μ ¼ 4ðϕÞ loops, we assume that at some UV scale Λ ≳ M the shift Vϕ above M contains the term dVϕ=d ln MS = 2 heavy symmetry of ϕ is broken only by its coupling to the Higgs. 2ð4πÞ þ, giving rise to a one-loop suppressed Vϕ : RGE running at one loop in the full theory above M contains the following two terms: 2 M g ϕ Λ VheavyðϕÞ ≃ S ln : ð7Þ 2 2 2 4 ϕ ð4πÞ2 dM ðϕÞ 4λ M dVϕ M ðϕÞ M h ¼ − HS ; ¼ h þ: ð4Þ d ln μ ð4πÞ2 d ln μ 2ð4πÞ2 Assuming ln Λ=M ∼ 1, the scale of new physics needed by In words: heavy new physics contributes to the Higgs mass the mechanism now is that contributes to the RGE running of the cosmological ϕ constant. As can be treated as a background field, in M ∼ 4πMh ∼ 1.5 TeV: ð8Þ view of its small coupling, we obtain the ϕ potential by iteratively solving the two RGE equations. Apart from a III. PRECARIOUS NATURALNESS: ϕ-independent constant, the dominant term is COSMOLOGICAL DYNAMICS 2 2 M M ðϕÞ Λ We here outline the main cosmological dynamics that VheavyðϕÞ ≃ h 2λ2 ln2 ; ð5Þ ϕ ð4πÞ4 HS M results in precarious naturalness. Possible issues will be addressed in the next section, finding bounds that can be 2 under the assumption that the RGE correction to Mh from satisfied. 2 new physics is much larger than Mh itself (otherwise, no In the primordial Universe, inflation creates patches with ϕ ðϕÞ Higgs mass naturalness problem is present [21]). This RGE different values of . The small Veff starts playing a role effect from boson loops has the desired sign.3 In this more only in the late Universe. In most patches ϕ is away from favorable scenario Vheavy is two-loop suppressed, so that it the maximum and slides down on a timescale much smaller ϕ 1 has the size needed by the mechanism under study for than the cosmological one, =H0. What is inside these patches undergoes a big crunch. Only in rare regions the field ϕ sits close enough to its maximum giving rise to a 3In order to see this, note that the boundary conditions for the heavyj ¼ 0 2ðϕÞj ¼ 2ðϕÞ metastable Universe. These regions, characterized by a running are Vϕ μ¼Λ , Mh μ≃M Mh . The former because at the scale Λ the shift symmetry is assumed to be broken small Higgs mass, get selected both anthropically and only by the coupling to the Higgs, the latter because the potential environmentally. Environmentally because the multiverse minimization is performed at the scale v ≈ M. volume, at late times, gets dominated by metastable regions

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4  2 2 that do not crunch quickly (see also [10]). Anthropically H =m if m ≲ H0 ϕ ≈ 0 ð Þ max MPl : 12 because observers can form only in these long-lived −m=H0 e H0=m if m ≳ H0 regions. The classical cosmological evolution of a homogeneous The range jϕj ≲ ϕ corresponds to a range of Higgs scalar ϕ around the top of its potential VðϕÞ¼−m2ϕ2=2, max pffiffiffiffiffi with m ¼ g= 8λ, is described by  2 2 H =m if m ≲ H0 j 2 − 2 j ≲ jϕj ∼ 0 2 M M 0 g gMPl ; ϕ ϕ h h −m=H0 d þ 3 d ¼ 2ϕ ð Þ e H0=m if m ≳ H0 2 H m ; 9 dt dt ð13Þ where H ¼ a=a_ is the Hubble rate. As long as H ≫ m the which is not larger than the observed small weak scale v if field slow rolls as 2 2 −29 m ≳ H0M =v ∼ 10 H0: ð14Þ Z  Pl t m2 ϕðtÞ ≈ ϕðt Þ exp d ln a : ð10Þ Equations (11) and (12) give the maximal range of ϕ such i 3H2 ti that its time evolution up to now leads to a change in 2 2 vacuum energy VϕðϕÞ¼−m ϕ =2 smaller than the Δ So, after the time t when HðtÞ becomes smaller observed vacuum energy Vmax. This field range contains than m, the field grows exponentially with timescale m: a range Vmax of vacuum energies larger than the time mðt−tÞ Δ ≲ ϕðtÞ ≈ ϕðtÞe . Even for m ≪ H0, the slower rolling variation Vmax if m H0: down leads to a deadly big crunch before the present epoch   2 H0 if the induced variation in Vϕ is larger than the present ¼ Δ ≲ 4 ð Þ Vmax Vmax v : 15 vacuum energy. m In order to compute the allowed range of ϕ, we impose Δ ∼ 2 2 that the change in VðϕÞ up to now (Hubble constant H0)is Inserting Vmax H0MPl, the maximal value of Vmax is Δ Oð1Þ achieved assuming the far-end lower range for m in smaller than some Vmax. Neglecting factors, this restricts the initial value of the field to be close to the Eq. (14). Such ultra-ultralight scalar allows also to tune 4 maximum within vacuum energies as large as the weak scale v down to 2 2 acceptable small values, of order H0MPl, as demanded by  anthropic considerations (see [24] and subsequent works) pffiffiffiffiffiffiffiffiffiffiffiffiffi 2 H0=m if m ≲ H0 jϕj ≲ ϕ ≡ ΔV : ð11Þ and provides a physical mechanism to realize such max max −m=H0 e =m if m ≳ H0 anthropic selection. The situation is illustrated in Fig. 2. 4 If the unknown constant V0 is smaller than about v , some jϕj ϕ field value within the range < max, allowed by stability The condition that cosmological evolution leads to an up to now, contains the observed small vacuum energy, and anthropically acceptable long-lived Universe with age is realized in some long-lived patch (left panel). Larger ≈1 Δ ∼ 2 2 =H0 corresponds to Vmax H0MPl, so that values of V0 only allow short-lived patches that roll down cosmologically fast (right panel), strengthening the usual 4Qualitatively different regions with a positive vacuum energy argument about anthropic selection of the cosmological V0 so large that they survive to ϕ sliding down are excluded only constant [24]. anthropically because of the large cosmological constant. Vol- umes can be estimated as IV. BOUNDS AND SIGNALS 3 2 ð3Þ M ð3Þ 1 M H ϕ V ≲ Pl V ≳ Pl 0 We next consider various bounds, showing that must collapsed 6 ; meta−stable 3 2 ; M H0 mΛ be lighter than ∼15H0. With this extra condition, the above discussion survives to all constraints. Then, we show that a ∼ as we now explain. Since collapsed patches are AdS we need to generic prediction of our framework is the presence of specify a measure: we calculate the volume as seen from inside [22,23], pessimistically assuming that the bottom of the potential sizable effects on the dark-energy equation of state. is at V ∼−M4 (more presumably is V ∼−Λ4). The volume of 3 metastable patches is conservatively taken as 1=H0 times the A. Bound from ϕ Δϕ small probability max= tot, where the latter is the total range of 2 We need to impose that quantum fluctuations (δϕ ≈ m ϕ. We pessimistically assume gΔϕtot ∼ Λ . Then, the volume Λ 3 per time 1=m) and de Sitter fluctuations (δϕ ∼ H0 per with light Higgs certainly dominates if

115002-4 RELAXING THE HIGGS MASS AND ITS VACUUM … PHYS. REV. D 101, 115002 (2020) condition is satisfied. For m ≳ H0 the desired range is C. Bound from thermal effects during the much smaller because the instability in Vϕ amplifies The field ϕ must be so weakly coupled that it negligibly fluctuations by the emt ∼ em=H0 factor, making them thermalizes. This is even more true at temperatures below classical (see, e.g., [25]). The field remains within the the weak scale, where its indirect couplings to SM particles allowed range if it is not much heavier than the Hubble are even more suppressed. scale At temperatures above the weak scale hjHj2i¼T2=6,so that the Lagrangian coupling −gϕjHj2=2 becomes a ther- MPl thermal 2 m ≲ H0 ln ∼ 100H0: ð16Þ mal contribution to the ϕ potential, Vϕ ≈−gT ϕ=12, H0 even if g is very small. Neglecting the nonthermal part of the potential, the thermal slope induces a drift of ϕ dominantly at the smallest temperature T ≳ v where B. Bound from inflationary dynamics min this term is present. In the slow-roll approximation We assumed that ϕ is near the top and nearly homo- geneous within the observable horizon. We now study if d2ϕ dϕ gT2 þ 3H ≈ such a region can be produced by inflation with Hubble dt2 dt 12 constant H . The possible issue is that, during inflation, Z 2 2 infl gT mM ϕ δϕ ≈ 2π ⇒ Δϕ ≈ ∼ Pl ð Þ the field undergoes de Sitter fluctuations Hinfl= thermal 2 d ln a 2 : 19 5 36 per e-fold. About N ≈ 50 e-folds of inflation are needed to H Tmin produce the observable Universe from one causal patch. Δϕ ϕ The rolled distance in field thermal is within the During this relevant inflationary period,pffiffiffiffi the field under- ϕ δϕ ≈ 2π metastability range max if goes the total fluctuation infl NHinfl= . Such infla- tionary fluctuation must be smaller than ϕ given in 2 max v Eq. (12). This condition is certainly satisfied if m ≲ H0, m ≲ H0 ln ∼ 60H0: ð20Þ ≪ ≳ H0MPl given that data demand Hinfl MPl.Form H0 we obtain the bound The similar and stronger bound from inflationary dynamics in Eq. (17) implies that the above condition is satisfied. ≲ pffiffiffiffiMPl ∼ 15 ∼ 1010 ð Þ m H0 ln H0 for Hinfl GeV: 17 Thereby we do not need to study the potentially successful NH infl but more complicated cosmological dynamics that takes Δϕ ϕ Unlike other relaxation mechanisms, the present one is place if, instead, thermal were larger than max. compatible with standard inflation at mildly sub-Planckian energy. D. Bound from variations of fundamental constants A similar but weaker bound is obtained by considering In the present scenario the weak scale v depends on ϕ, ϕ_ fluctuationspffiffiffiffiffiffiffiffi in the velocity of the fieldpffiffiffiffi at inflation end, and its variation induces variations of other fundamental δϕ_ ∼ 3 2 2 2π infl = Hinfl= . There is no N enhancement constants. Bounds on the time variation of fundamental because a free classical field satisfies ϕ̈þ 3ða=a_ Þϕ_ ¼ 0 constants have been derived from cosmological observa- so that its velocity redshifts as ϕ_ ∝ 1=a3: earlier infla- tions. For our purposes, the strongest bound is on the tionary fluctuations in δϕ_ get diluted and can be variation of the ratio between the and proton mass infl ¼ neglected. For the same reason an initial velocity does r me=mp, proportional to v. The bound is [26] not lead to a large field variation in the subsequent d ln r 1 postinflationary evolution. The total postinflationary shift ≲ 10−7 at t ∼ : ð21Þ in the field is dominated by a few Hubble times just after d ln t H0 ≈ inflation, during which the Hubble rate still is Hinfl because of conservation of energy: These bounds are largely subdominant with respect to bounds on variations of the cosmological constant. Indeed a Z 7 δϕ_ change in the electron mass by one part in 10 implies a Δϕ ¼ dtϕ_ ≈ infl ≈ H : ð18Þ ≈1030 H infl huge variation in the cosmological constant. This infl presumably happens in any theory (with the possible pffiffiffiffi δϕ ≈ exception of speculative theories with self-cancellation This is smaller than infl by a factor N and thereby gives a weaker bound than Eq. (17). of the cosmological constant) [27]. In particular, this conclusion holds in the theory under consideration, where

5Notice that the shift symmetry also suppresses nonminimal ϕ ϕ 2 ϕ ðϕÞ dlnr ¼ dlnv dln ∼ m m ≲ mMPl ≪ 10−7 ð Þ couplings of to gravity f R, which would complicate its × 2 × 2 2 ; 22 cosmological evolution in the early Universe. dlnt dlnϕ dlnt v H v

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Destabilized by inflation

Unstable o fthe of top

Meta stable potential

2 2 FIG. 3. The light scalar potential can be approximated, around its top, as Vϕ ≃−m ϕ =2 so that the model depends on m (vertical axis) and on the field value ϕ (horizontal axis). Regions shaded in red with m ≳ 15H0 are excluded because inflationary perturbations are too large, in usual models of high-scale inflation. Regions in grey are excluded because a too fast rolling prevents a long-lived Universe. The slightly larger hatched region is observationally excluded because the vacuum energy deviates too much from a cosmological constant. −29 Green regions are anthropically acceptable and extend down to m ≳ 10 H0.

6 having used the allowed range of ϕ from Eq. (12).In regimes m ≲ H0 and m ≳ H0. In order to discuss the conclusion, bounds from variations of fundamental con- expected value of w0 we need to consider separately the stants are negligible. two cases: (1) If m ≲ H0 the vacuum energy significantly varies in jϕj ϕ E. Signal from dark-energy equation of state the range < max. The anthropic bound on the vacuum energy [24] further restricts ϕ to a narrower As argued above, the only fundamental parameter that region, as illustrated in Fig. 2(a).Thevalueofϕ=ϕ can undergo an observable cosmological evolution is the max is fixed but unknown, and determines w0 as in cosmological “constant.” Such effect is usually parame- Eq. (23). trized by deviations from −1 of its equation-of-state (2) If m ≳ H0 the vacuum energy does not significantly parameter jϕj ϕ vary in the range < max. The cosmological 2 2 pϕ Vϕ − Kϕ 1 − ϕ ϕ constant does not give extra restrictions on ϕ, ≡ ¼ − ≈− = max ð Þ w 2 2 ; 23 that is not expected to be significantly smaller ρϕ Vϕ þ Kϕ 1 þ ϕ ϕ = max ϕ than max. _ 2 where Kϕ ¼ ϕ =2.Ifm ≳ H0 the red-shift of Kϕ is non- In the second case (and, perhaps, in the first case), we expect a ϕ −ϕ ≲ negligible. Despite this, the latter equation approximately flat of in the range max ϕ ≲ ϕ ϕ holds because, independently from other model parameters, max. Therefore, we expect that the field is sliding ϕ ϕ down the potential today, thus having observable effects in the maximal field value max above which rolls down catastrophically fast corresponds to order unity deviations of the dark-energy equation of state. The experimental bound w from −1. The quadratic dependence on ϕ is robust and w0 ¼ −1.03 0.03 [28] excludes an order unity fraction of 3σ arises because VϕðϕÞ can be approximated as a quadratic the parameterpffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi space. Avalue of w within its range arises in Taylor expansion around the maximum of the potential afraction≈ 3 × 0.03 ∼ 1=3 of the parameter space. A more at ϕ ¼ 0. precise determination of the probability that a long-lived The situation is illustrated in Fig. 3. For any given patch is compatible with bounds on w requires a more jϕj ϕ value of m, patches with > max, corresponding to j þ 1j ≳ 1 w , are excluded because short-lived (grey 6 regions). Among the anthropically acceptable patches, an We stress here that, although at first sight Fig. 3 can naively indicate that the mechanism is not effective for m ≳ H0, this is not order one fraction (in hatched green) has now been the case, since successful long-lived patches would be generated excluded by cosmological bounds. The remaining part in any case somewhere in the Universe. For m ≳ H0 they are (in green) is fully acceptable. This is true for both the simply more sparse.

115002-6 RELAXING THE HIGGS MASS AND ITS VACUUM … PHYS. REV. D 101, 115002 (2020) detailed computation, that goes beyond the scope of where T0 is the present temperature. The lower bound arises this work. demanding that theweak scale is generated. The upper bound In conclusion, a dynamical dark energy is a generic arises from stability against inflationary fluctuations, con- prediction, with an observable deviation of w0 from −1 in sidering the usual scenario of high-scale inflation. an Oð1Þ fraction of the long-lived patches. If m is smaller than H0, the stability range of ϕ covers significantly different vacuum energies.7 The smallest m F. Collider signals allows for variations in the vacuum energy density up to 4 The scenario under consideration makes the Higgs electroweak-scale ones, v . Therefore, within this range, the lighter than new physics by a one-loop or two-loop factor, model also allows for anthropic selection of the vacuum so that new physics is expected around the scale indicated energy, in addition to selecting a Higgs mass lighter than new in Eq. (8) or Eq. (6), respectively. Some unspecified new physics. physics around this scale is actually needed by the scenario. The main novelty of this scenario is that the Universe is New physics could be some full solution to the hierarchy near-critical, with ϕ being close to the maximum of its problem, such as supersymmetry. The collider signals are potential, rather than in a minimum. As such, naturalness is well known, having been studied when natural new physics precarious: ϕ will necessarily roll down in the future and the at the weak scale was a viable possibility. In the one-loop Universe will collapse into a big crunch. We expect that the case, new physics coupled to the Higgs is expected around Universe is already rolling now, with an Oð1Þ probability 1.5 TeV. This is typically consistent with current constraints to have observable effects in the dark-energy equation of state. from collider and precision data, as long as new physics is Precarious naturalness introduces only a little hierarchy not colored and does not violate flavor nor CP.For between the new-physics scale and the Higgs mass. In the example, in this mass range the thermal relic abundance most favorable case, where new physics couples to of neutral stable particles with electroweak interactions can ϕ only via the Higgs, a two-loop hierarchy is generated, 2 reproduce the cosmological abundance. M ∼ ð4πÞ Mh ∼ 20 TeV. More in general, the light scalar ϕ couples to the Higgs and to new physics, and a one-loop V. CONCLUSIONS hierarchy Mh ∼ 4πMh ∼ 1.5 TeV is generated. Some special new physics, such as supersymmetry, We considered an ultralight scalar ϕ, tinily coupled to the allows for a full solution to the hierarchy problem. Higgs and to heavier new physics. Its potential can easily have a maximum at field values such that the Higgs is However no new physics has been observed at the weak significantly lighter than the new-physics scale M, since the scale. By generating a little hierarchy, the present scenario alleviates this problem, establishing a connection between Higgs contributes to the ϕ potential Vϕ at tree level, while collider and dark-energy experiments. These fields are in many models heavier new physics only contributes at presently in a precarious state, given that no new physics one- or two-loop level. Inflation creates different patches of was discovered despite significant efforts. Precarious natu- the Universe with different values of ϕ. Patches where ϕ is sufficiently close to the maximum survive for a long, ralness would guarantee future discoveries, at the price of cosmological, time. Patches with ϕ farther away quickly making the Universe itself precarious. collapse into a big crunch, as seen from inside. Therefore, a small Higgs mass is selected both anthropically (because ACKNOWLEDGMENTS observers can form only in long-lived regions) and envi- ronmentally (because the volume of the multiverse is This work was supported by the ERC Grant No. 669668 dominated by long-lived regions). NEO-NAT. We thank Alfredo Urbano and Brando Bellazzini 2 2 for discussions. Approximating the ϕ potential as Vϕ ≃−m ϕ =2, the allowed range of m is

2 7 10−29 ∼ T0 ≲ m ≲ 10 ð Þ In this case the ϕ field excursion is largely super-Planckian, as 2 24 v H0 it is typically the case in models of relaxation.

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